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**A First Course in Differential Equations, Modeling, and Simulation** shows how differential equations arise from applying basic physical principles and experimental observations to engineering systems. Avoiding overly theoretical explanations, the textbook also discusses classical and Laplace transform methods for obtaining the analytical solution of differential equations. In addition, the authors explain how to solve sets of differential equations where analytical solutions cannot easily be obtained.

Incorporating valuable suggestions from mathematicians and mathematics professors, the **Second Edition**:

- Expands the chapter on classical solutions of ordinary linear differential equations to include additional methods
- Increases coverage of response of first- and second-order systems to a full, stand-alone chapter to emphasize its importance
- Includes new examples of applications related to chemical reactions, environmental engineering, biomedical engineering, and biotechnology
- Contains new exercises that can be used as projects and answers to many of the end-of-chapter problems
- Features new end-of-chapter problems and updates throughout

Thus, **A First Course in Differential Equations, Modeling, and Simulation, Second Edition** provides students with a practical understanding of how to apply differential equations in modern engineering and science.

**Introduction**

An Introductory Example

Differential Equations

Modeling

Forcing Functions

Book Objectives

Summary

Objects in a Gravitational Field

An Example

Antidifferentiation: Technique for Solving First-Order Ordinary Differential Equations

Back to Section 2.1

Another Example

Separation of Variables: Technique for Solving First-Order Ordinary Differential Equations

Back to Section 2.4

Equations, Unknowns, and Degrees of Freedom

Partial Fraction Expansion

Summary

Classical Solutions of Ordinary Linear Differential Equations

Examples of Differential Equations

Definition of a Linear Differential Equation

Integrating Factor Method

Solution of Homogeneous Differential Equations

Solution of Nonhomogeneous Differential Equations

Variation of Parameters

Handling Nonlinearities and Variable Coefficients

Transient and Final Responses

Summary

Laplace Transforms

Definition of the Laplace Transform

Properties and Theorems of the Laplace Transform

Solution of Differential Equations Using Laplace Transform

Transfer Functions

Algebraic Manipulations Using Laplace Transforms

Deviation Variables

Summary

Response of First- and Second-Order Systems

First-Order Systems

Second-Order Systems

Examples

Some Concluding Remarks

Summary

Mechanical Systems: Translational

Mechanical Law, System Components, and Forces

Types of Systems

D’Alembert’s Principle and Free Body Diagrams

Examples

Vertical Systems

Summary

Mechanical Systems: Rotational

Mechanical Law, Moment of Inertia, and Torque

Torsion Springs

Rotational Damping

Gears

Systems with Rotational and Translational Elements

Summary

Mass Balances

Conservation of Mass

Flow Rates and Concentrations

Elements and Experimental Facts

Examples

Expressions for Mass Transport and Chemical Reactions

Additional Examples

Application to Bioengineering Processes

Final Comments

Summary

References

Thermal Systems

Conservation of Energy

Modes of Heat Transfer

Conduction

Convection

Conduction and Convection in Series

Accumulated or Stored Energy

Some Examples

Heat Transfer in a Flow System

Thermal Effects in a Reactive System

Boundary Value Problems in Heat Transfer

Summary

Electrical Systems

Some Definitions and Conventions

Electrical Laws, Components, and Initial Conditions

Examples of Electrical Circuits

Additional Examples

Energy and Power

RC Circuits as Filters

Summary

Numerical Simulation

Numerical Solution of Differential Equations

Euler’s Method for First-Order Ordinary Differential Equations

Euler’s Method for Second-Order Ordinary Differential Equations

Step Size

More Sophisticated Methods

Representation of Differential Equations by Block Diagrams

Additional Examples

Summary

Reference

Answers to Selected Problems

**A First Course in Differential Equations, Modeling, and Simulation** shows how differential equations arise from applying basic physical principles and experimental observations to engineering systems. Avoiding overly theoretical explanations, the textbook also discusses classical and Laplace transform methods for obtaining the analytical solution of differential equations. In addition, the authors explain how to solve sets of differential equations where analytical solutions cannot easily be obtained.

Incorporating valuable suggestions from mathematicians and mathematics professors, the **Second Edition**:

- Expands the chapter on classical solutions of ordinary linear differential equations to include additional methods
- Increases coverage of response of first- and second-order systems to a full, stand-alone chapter to emphasize its importance
- Includes new examples of applications related to chemical reactions, environmental engineering, biomedical engineering, and biotechnology
- Contains new exercises that can be used as projects and answers to many of the end-of-chapter problems
- Features new end-of-chapter problems and updates throughout

Thus, **A First Course in Differential Equations, Modeling, and Simulation, Second Edition** provides students with a practical understanding of how to apply differential equations in modern engineering and science.

**Introduction**

An Introductory Example

Differential Equations

Modeling

Forcing Functions

Book Objectives

Summary

Objects in a Gravitational Field

An Example

Antidifferentiation: Technique for Solving First-Order Ordinary Differential Equations

Back to Section 2.1

Another Example

Separation of Variables: Technique for Solving First-Order Ordinary Differential Equations

Back to Section 2.4

Equations, Unknowns, and Degrees of Freedom

Partial Fraction Expansion

Summary

Classical Solutions of Ordinary Linear Differential Equations

Examples of Differential Equations

Definition of a Linear Differential Equation

Integrating Factor Method

Solution of Homogeneous Differential Equations

Solution of Nonhomogeneous Differential Equations

Variation of Parameters

Handling Nonlinearities and Variable Coefficients

Transient and Final Responses

Summary

Laplace Transforms

Definition of the Laplace Transform

Properties and Theorems of the Laplace Transform

Solution of Differential Equations Using Laplace Transform

Transfer Functions

Algebraic Manipulations Using Laplace Transforms

Deviation Variables

Summary

Response of First- and Second-Order Systems

First-Order Systems

Second-Order Systems

Examples

Some Concluding Remarks

Summary

Mechanical Systems: Translational

Mechanical Law, System Components, and Forces

Types of Systems

D’Alembert’s Principle and Free Body Diagrams

Examples

Vertical Systems

Summary

Mechanical Systems: Rotational

Mechanical Law, Moment of Inertia, and Torque

Torsion Springs

Rotational Damping

Gears

Systems with Rotational and Translational Elements

Summary

Mass Balances

Conservation of Mass

Flow Rates and Concentrations

Elements and Experimental Facts

Examples

Expressions for Mass Transport and Chemical Reactions

Additional Examples

Application to Bioengineering Processes

Final Comments

Summary

References

Thermal Systems

Conservation of Energy

Modes of Heat Transfer

Conduction

Convection

Conduction and Convection in Series

Accumulated or Stored Energy

Some Examples

Heat Transfer in a Flow System

Thermal Effects in a Reactive System

Boundary Value Problems in Heat Transfer

Summary

Electrical Systems

Some Definitions and Conventions

Electrical Laws, Components, and Initial Conditions

Examples of Electrical Circuits

Additional Examples

Energy and Power

RC Circuits as Filters

Summary

Numerical Simulation

Numerical Solution of Differential Equations

Euler’s Method for First-Order Ordinary Differential Equations

Euler’s Method for Second-Order Ordinary Differential Equations

Step Size

More Sophisticated Methods

Representation of Differential Equations by Block Diagrams

Additional Examples

Summary

Reference

Answers to Selected Problems

**A First Course in Differential Equations, Modeling, and Simulation** shows how differential equations arise from applying basic physical principles and experimental observations to engineering systems. Avoiding overly theoretical explanations, the textbook also discusses classical and Laplace transform methods for obtaining the analytical solution of differential equations. In addition, the authors explain how to solve sets of differential equations where analytical solutions cannot easily be obtained.

Incorporating valuable suggestions from mathematicians and mathematics professors, the **Second Edition**:

- Expands the chapter on classical solutions of ordinary linear differential equations to include additional methods
- Increases coverage of response of first- and second-order systems to a full, stand-alone chapter to emphasize its importance
- Includes new examples of applications related to chemical reactions, environmental engineering, biomedical engineering, and biotechnology
- Contains new exercises that can be used as projects and answers to many of the end-of-chapter problems
- Features new end-of-chapter problems and updates throughout

Thus, **A First Course in Differential Equations, Modeling, and Simulation, Second Edition** provides students with a practical understanding of how to apply differential equations in modern engineering and science.

**Introduction**

An Introductory Example

Differential Equations

Modeling

Forcing Functions

Book Objectives

Summary

Objects in a Gravitational Field

An Example

Antidifferentiation: Technique for Solving First-Order Ordinary Differential Equations

Back to Section 2.1

Another Example

Separation of Variables: Technique for Solving First-Order Ordinary Differential Equations

Back to Section 2.4

Equations, Unknowns, and Degrees of Freedom

Partial Fraction Expansion

Summary

Classical Solutions of Ordinary Linear Differential Equations

Examples of Differential Equations

Definition of a Linear Differential Equation

Integrating Factor Method

Solution of Homogeneous Differential Equations

Solution of Nonhomogeneous Differential Equations

Variation of Parameters

Handling Nonlinearities and Variable Coefficients

Transient and Final Responses

Summary

Laplace Transforms

Definition of the Laplace Transform

Properties and Theorems of the Laplace Transform

Solution of Differential Equations Using Laplace Transform

Transfer Functions

Algebraic Manipulations Using Laplace Transforms

Deviation Variables

Summary

Response of First- and Second-Order Systems

First-Order Systems

Second-Order Systems

Examples

Some Concluding Remarks

Summary

Mechanical Systems: Translational

Mechanical Law, System Components, and Forces

Types of Systems

D’Alembert’s Principle and Free Body Diagrams

Examples

Vertical Systems

Summary

Mechanical Systems: Rotational

Mechanical Law, Moment of Inertia, and Torque

Torsion Springs

Rotational Damping

Gears

Systems with Rotational and Translational Elements

Summary

Mass Balances

Conservation of Mass

Flow Rates and Concentrations

Elements and Experimental Facts

Examples

Expressions for Mass Transport and Chemical Reactions

Additional Examples

Application to Bioengineering Processes

Final Comments

Summary

References

Thermal Systems

Conservation of Energy

Modes of Heat Transfer

Conduction

Convection

Conduction and Convection in Series

Accumulated or Stored Energy

Some Examples

Heat Transfer in a Flow System

Thermal Effects in a Reactive System

Boundary Value Problems in Heat Transfer

Summary

Electrical Systems

Some Definitions and Conventions

Electrical Laws, Components, and Initial Conditions

Examples of Electrical Circuits

Additional Examples

Energy and Power

RC Circuits as Filters

Summary

Numerical Simulation

Numerical Solution of Differential Equations

Euler’s Method for First-Order Ordinary Differential Equations

Euler’s Method for Second-Order Ordinary Differential Equations

Step Size

More Sophisticated Methods

Representation of Differential Equations by Block Diagrams

Additional Examples

Summary

Reference

Answers to Selected Problems

**A First Course in Differential Equations, Modeling, and Simulation** shows how differential equations arise from applying basic physical principles and experimental observations to engineering systems. Avoiding overly theoretical explanations, the textbook also discusses classical and Laplace transform methods for obtaining the analytical solution of differential equations. In addition, the authors explain how to solve sets of differential equations where analytical solutions cannot easily be obtained.

**Second Edition**:

- Features new end-of-chapter problems and updates throughout

**A First Course in Differential Equations, Modeling, and Simulation, Second Edition** provides students with a practical understanding of how to apply differential equations in modern engineering and science.

**Introduction**

An Introductory Example

Differential Equations

Modeling

Forcing Functions

Book Objectives

Summary

Objects in a Gravitational Field

An Example

Antidifferentiation: Technique for Solving First-Order Ordinary Differential Equations

Back to Section 2.1

Another Example

Separation of Variables: Technique for Solving First-Order Ordinary Differential Equations

Back to Section 2.4

Equations, Unknowns, and Degrees of Freedom

Partial Fraction Expansion

Summary

Classical Solutions of Ordinary Linear Differential Equations

Examples of Differential Equations

Definition of a Linear Differential Equation

Integrating Factor Method

Solution of Homogeneous Differential Equations

Solution of Nonhomogeneous Differential Equations

Variation of Parameters

Handling Nonlinearities and Variable Coefficients

Transient and Final Responses

Summary

Laplace Transforms

Definition of the Laplace Transform

Properties and Theorems of the Laplace Transform

Solution of Differential Equations Using Laplace Transform

Transfer Functions

Algebraic Manipulations Using Laplace Transforms

Deviation Variables

Summary

Response of First- and Second-Order Systems

First-Order Systems

Second-Order Systems

Examples

Some Concluding Remarks

Summary

Mechanical Systems: Translational

Mechanical Law, System Components, and Forces

Types of Systems

D’Alembert’s Principle and Free Body Diagrams

Examples

Vertical Systems

Summary

Mechanical Systems: Rotational

Mechanical Law, Moment of Inertia, and Torque

Torsion Springs

Rotational Damping

Gears

Systems with Rotational and Translational Elements

Summary

Mass Balances

Conservation of Mass

Flow Rates and Concentrations

Elements and Experimental Facts

Examples

Expressions for Mass Transport and Chemical Reactions

Additional Examples

Application to Bioengineering Processes

Final Comments

Summary

References

Thermal Systems

Conservation of Energy

Modes of Heat Transfer

Conduction

Convection

Conduction and Convection in Series

Accumulated or Stored Energy

Some Examples

Heat Transfer in a Flow System

Thermal Effects in a Reactive System

Boundary Value Problems in Heat Transfer

Summary

Electrical Systems

Some Definitions and Conventions

Electrical Laws, Components, and Initial Conditions

Examples of Electrical Circuits

Additional Examples

Energy and Power

RC Circuits as Filters

Summary

Numerical Simulation

Numerical Solution of Differential Equations

Euler’s Method for First-Order Ordinary Differential Equations

Euler’s Method for Second-Order Ordinary Differential Equations

Step Size

More Sophisticated Methods

Representation of Differential Equations by Block Diagrams

Additional Examples

Summary

Reference

Answers to Selected Problems

**A First Course in Differential Equations, Modeling, and Simulation** shows how differential equations arise from applying basic physical principles and experimental observations to engineering systems. Avoiding overly theoretical explanations, the textbook also discusses classical and Laplace transform methods for obtaining the analytical solution of differential equations. In addition, the authors explain how to solve sets of differential equations where analytical solutions cannot easily be obtained.

**Second Edition**:

- Features new end-of-chapter problems and updates throughout

**A First Course in Differential Equations, Modeling, and Simulation, Second Edition** provides students with a practical understanding of how to apply differential equations in modern engineering and science.

**Introduction**

An Introductory Example

Differential Equations

Modeling

Forcing Functions

Book Objectives

Summary

Objects in a Gravitational Field

An Example

Antidifferentiation: Technique for Solving First-Order Ordinary Differential Equations

Back to Section 2.1

Another Example

Separation of Variables: Technique for Solving First-Order Ordinary Differential Equations

Back to Section 2.4

Equations, Unknowns, and Degrees of Freedom

Partial Fraction Expansion

Summary

Classical Solutions of Ordinary Linear Differential Equations

Examples of Differential Equations

Definition of a Linear Differential Equation

Integrating Factor Method

Solution of Homogeneous Differential Equations

Solution of Nonhomogeneous Differential Equations

Variation of Parameters

Handling Nonlinearities and Variable Coefficients

Transient and Final Responses

Summary

Laplace Transforms

Definition of the Laplace Transform

Properties and Theorems of the Laplace Transform

Solution of Differential Equations Using Laplace Transform

Transfer Functions

Algebraic Manipulations Using Laplace Transforms

Deviation Variables

Summary

Response of First- and Second-Order Systems

First-Order Systems

Second-Order Systems

Examples

Some Concluding Remarks

Summary

Mechanical Systems: Translational

Mechanical Law, System Components, and Forces

Types of Systems

D’Alembert’s Principle and Free Body Diagrams

Examples

Vertical Systems

Summary

Mechanical Systems: Rotational

Mechanical Law, Moment of Inertia, and Torque

Torsion Springs

Rotational Damping

Gears

Systems with Rotational and Translational Elements

Summary

Mass Balances

Conservation of Mass

Flow Rates and Concentrations

Elements and Experimental Facts

Examples

Expressions for Mass Transport and Chemical Reactions

Additional Examples

Application to Bioengineering Processes

Final Comments

Summary

References

Thermal Systems

Conservation of Energy

Modes of Heat Transfer

Conduction

Convection

Conduction and Convection in Series

Accumulated or Stored Energy

Some Examples

Heat Transfer in a Flow System

Thermal Effects in a Reactive System

Boundary Value Problems in Heat Transfer

Summary

Electrical Systems

Some Definitions and Conventions

Electrical Laws, Components, and Initial Conditions

Examples of Electrical Circuits

Additional Examples

Energy and Power

RC Circuits as Filters

Summary

Numerical Simulation

Numerical Solution of Differential Equations

Euler’s Method for First-Order Ordinary Differential Equations

Euler’s Method for Second-Order Ordinary Differential Equations

Step Size

More Sophisticated Methods

Representation of Differential Equations by Block Diagrams

Additional Examples

Summary

Reference

Answers to Selected Problems

**A First Course in Differential Equations, Modeling, and Simulation** shows how differential equations arise from applying basic physical principles and experimental observations to engineering systems. Avoiding overly theoretical explanations, the textbook also discusses classical and Laplace transform methods for obtaining the analytical solution of differential equations. In addition, the authors explain how to solve sets of differential equations where analytical solutions cannot easily be obtained.

**Second Edition**:

- Features new end-of-chapter problems and updates throughout

**A First Course in Differential Equations, Modeling, and Simulation, Second Edition** provides students with a practical understanding of how to apply differential equations in modern engineering and science.

**Introduction**

An Introductory Example

Differential Equations

Modeling

Forcing Functions

Book Objectives

Summary

Objects in a Gravitational Field

An Example

Antidifferentiation: Technique for Solving First-Order Ordinary Differential Equations

Back to Section 2.1

Another Example

Separation of Variables: Technique for Solving First-Order Ordinary Differential Equations

Back to Section 2.4

Equations, Unknowns, and Degrees of Freedom

Partial Fraction Expansion

Summary

Classical Solutions of Ordinary Linear Differential Equations

Examples of Differential Equations

Definition of a Linear Differential Equation

Integrating Factor Method

Solution of Homogeneous Differential Equations

Solution of Nonhomogeneous Differential Equations

Variation of Parameters

Handling Nonlinearities and Variable Coefficients

Transient and Final Responses

Summary

Laplace Transforms

Definition of the Laplace Transform

Properties and Theorems of the Laplace Transform

Solution of Differential Equations Using Laplace Transform

Transfer Functions

Algebraic Manipulations Using Laplace Transforms

Deviation Variables

Summary

Response of First- and Second-Order Systems

First-Order Systems

Second-Order Systems

Examples

Some Concluding Remarks

Summary

Mechanical Systems: Translational

Mechanical Law, System Components, and Forces

Types of Systems

D’Alembert’s Principle and Free Body Diagrams

Examples

Vertical Systems

Summary

Mechanical Systems: Rotational

Mechanical Law, Moment of Inertia, and Torque

Torsion Springs

Rotational Damping

Gears

Systems with Rotational and Translational Elements

Summary

Mass Balances

Conservation of Mass

Flow Rates and Concentrations

Elements and Experimental Facts

Examples

Expressions for Mass Transport and Chemical Reactions

Additional Examples

Application to Bioengineering Processes

Final Comments

Summary

References

Thermal Systems

Conservation of Energy

Modes of Heat Transfer

Conduction

Convection

Conduction and Convection in Series

Accumulated or Stored Energy

Some Examples

Heat Transfer in a Flow System

Thermal Effects in a Reactive System

Boundary Value Problems in Heat Transfer

Summary

Electrical Systems

Some Definitions and Conventions

Electrical Laws, Components, and Initial Conditions

Examples of Electrical Circuits

Additional Examples

Energy and Power

RC Circuits as Filters

Summary

Numerical Simulation

Numerical Solution of Differential Equations

Euler’s Method for First-Order Ordinary Differential Equations

Euler’s Method for Second-Order Ordinary Differential Equations

Step Size

More Sophisticated Methods

Representation of Differential Equations by Block Diagrams

Additional Examples

Summary

Reference

Answers to Selected Problems